In a food processor or blender, in batches if necessary, combine all of the ingredients and puree until smooth. Taste and add more apple juice concentrate or mint if needed to balance the ripeness of the melon. If you have an ice cream maker, freeze the mixture according to the manufacturer's instructions, or transfer the mixture to an airtight container, place in the freezer, and freeze for several hours or until firm. One hour before serving, process the mixture in the food processor again, return it to the airtight container, and refreeze until firm. Serve in chilled glasses and, if desired, add a garnish of fresh fruit and a sprig of mint.